Advances in Operator Theory

Dominated orthogonally additive operators in lattice-normed spaces

Nariman Abasov and Marat Pliev

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‎In this paper, we introduce a new class of operators in lattice-normed spaces‎. ‎We say that an orthogonally additive operator $T$ from a lattice-normed space $(V,E)$ to a lattice-normed space‎ ‎$(W,F)$ is dominated‎, ‎if there exists a positive orthogonally additive operator $S$ from $E$ to $F$ such that $\vert Tx \vert \leq S \vert x \vert$ for any element $x$ of $(V,E)$‎. ‎We show that under some mild‎ ‎conditions‎, ‎a dominated orthogonally additive operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated orthogonally additive operator‎. ‎In the last part of the‎ ‎paper we consider laterally-to-order continuous operators‎. ‎We prove that a dominated orthogonally additive operator is laterally-to-order continuous if and only if the same is its exact dominant‎.

Article information

Adv. Oper. Theory, Volume 4, Number 1 (2019), 251-264.

Received: 27 April 2018
Accepted: 2 August 2018
First available in Project Euclid: 20 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]
Secondary: 47H99: None of the above, but in this section

Lattice-normed space‎ ‎vector lattice‎‎ ‎orthogonally additive operator‎ ‎dominated $\mathcal{P}$-operator ‎exact dominant‎ ‎‎laterally-to-order continuous operator‎


Abasov, Nariman; Pliev, Marat. Dominated orthogonally additive operators in lattice-normed spaces. Adv. Oper. Theory 4 (2019), no. 1, 251--264. doi:10.15352/aot.1804-1354.

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  • N. Abasov and M. Pliev, On extensions of some nonlinear maps in vector lattices, J. Math. Anal. Appl. 455 (2017), 516–527.
  • C. D. Aliprantis and O. Burkinshaw, Positive operators, Reprint of the 1985 original. Springer, Dordrecht, 2006.
  • A. Aydin, E. Yu Emelyanov, N. Erkursun Ozcan, and M. A. A. Marabeh, Compact-like operators in lattice-normed spaces, Indag. Math. 29 (2018), no. 2, 633–656.
  • J. Batt Nonlinear integral operators on $C(S,E)$, Studia Math. 48 (1973), 145–177.
  • J. Diestel and J. J. Uhl, Vector measures, With a foreword by B. J. Pettis. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
  • X. Fang and M. Pliev, Narrow orthgonally additive operators in lattice-normed spaces, Siberian Math. J. 58 (2017), no. 1, 134–141.
  • W. A. Feldman, Lattice preserving maps on lattices of continuous functions, J. Math. Anal. Appl. 404 (2013), 310–316.
  • W. A. Feldman, A characterization of non-linear maps satisfying orthogonality properties, Positivity 21 (2017), no. 1, 85–97.
  • W. A. Feldman and S. Pramod, A characterization of positively decomposable non-linear maps between Banach lattices, Positivity 12 (2008), 495–502.
  • N. Friedman and M. Katz A representaion theorem for additive functionals, Arch. Rational Mech. Anal. 21 (1966), no. 1, 49–57.
  • H. I. Gumenchuk, On the sum of narrow and finite-rank orthogonally additive operators, Ukrainian Math. J. 67 (2016), no. 12, 1831–1837.
  • A. G. Kusraev, Dominated operators, Translated from the 1999 Russian original by the author. Translation edited and with a foreword by S. Kutateladze. Mathematics and its Applications, 519. Kluwer Academic Publishers, Dordrecht, 2000.
  • M. Marcus and V. Mizel Representaion theorem for nonlinear disjointly additive functionals and operators on Sobolev spaces, Trans. Amer. Math. Soc. 226 (1977), 1–45.
  • M. Marcus and V. Mizel Extension theorem of Hahn-Banach type for nonlinear disjontly additive functionals and operators in Lebesgue spaces, J. Funct. Anal. 24 (1977), 303–335.
  • J. M. Mazón and S. Segura de León, Order bounded orthogonally additive operators, Rev. Roumane Math. Pures Appl. 35 (1990), no. 4, 329–353.
  • J. M. Mazón and S. Segura de León, Uryson operators, Rev. Roumane Math. Pures Appl. 35 (1990), no. 5, 431–449.
  • V. Orlov, M. Pliev, and D. Rode Domination problem for AM-compact abstract Uryson operators, Arch. Math. (Basel) 107 (2016), no. 5, 543–552.
  • M. Pliev Domination problem for narrow orthogonally additive operators, Positivity 21 (2017), no. 1, 23–33.
  • M. Pliev and K. Ramdane Order unbounded orthogonally additive operators in vector lattices, Mediterr. J. Math. 15 (2018), no. 2, Art. 55, 20 pp.
  • S. Segura de León, Bukhvalov type characterization of Urysohn operators, Studia Math. 99 (1991), no. 3, 199–220.